Derousseau's Generalization of the Malfatti circles

Equilateral Triangle

\(a:b:c=1:1:1\), \(A=60\degree\), \(B=60\degree\), \(C=60\degree\).


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6(220)

Malfatti circles

6 (220)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \\ P_{\mathbf{6}}&{}\approx{}&0.498714618890&{}:{}&0.498714618890&{}:{}&0.002570762220&, \\ P^-_{\mathbf{6}}&{}\approx{}&0.609378523886&{}:{}&0.609378523886&{}:{}&-0.218757047773&, \\ P^+_{\mathbf{6}}&{}\approx{}&0.451387728771&{}:{}&0.451387728771&{}:{}&0.097224542457&, \\ Q_{\mathbf{6}}&{}\approx{}&0.518617531751&{}:{}&0.518617531751&{}:{}&-0.037235063502&, \\ I^\prime_{\mathbf{6}}&{}\approx{}&0.482672825160&{}:{}&0.482672825160&{}:{}&0.034654349680&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{6}}\)
\(P^-_{\mathbf{6}}\)
\(P^+_{\mathbf{6}}\)
\(Q_{\mathbf{6}}\)
\(I^\prime_{\mathbf{6}}\)
6 (220)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6}}&{}\approx{}&0.577350269190&{}:{}&0.211324865405&{}:{}&0.211324865405&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.211324865405&{}:{}&0.577350269190&{}:{}&0.211324865405&,\\C^\prime_{\mathbf{6}}&{}\approx{}&0.981125224325&{}:{}&0.981125224325&{}:{}&-0.962250448649&, \\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.347748149397&{}:{}&0.648906880894&{}:{}&0.003344969709&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.648906880894&{}:{}&0.347748149397&{}:{}&0.003344969709&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.394337567297&{}:{}&0.394337567297&{}:{}&0.211324865405&, \\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.363466037631&{}:{}&0.993007681634&{}:{}&-0.356473719264&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.993007681634&{}:{}&0.363466037631&{}:{}&-0.356473719264&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.422649730810&{}:{}&0.422649730810&{}:{}&0.154700538379&, \\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.077350269190&{}:{}&-0.077350269190&,\\B^*_{\mathbf{6}}&{}\approx{}&1.077350269190&{}:{}&0.000000000000&{}:{}&-0.077350269190&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6}}}{B^\prime_{\mathbf{6}}}{C^\prime_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6}}}{B^{\prime\prime}_{\mathbf{6}}}{C^{\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6}}}{B^{\prime\prime\prime}_{\mathbf{6}}}{C^{\prime\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^*_{\mathbf{6}}}{B^*_{\mathbf{6}}}{C^*_{\mathbf{6}}}\)
6 (220)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6}}}}&{}\approx{}&0.633974596216&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6}}}}&{}\approx{}&0.633974596216&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6}}}}&{}\approx{}&2.943375672974&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.577350269190&{}:{}&0.211324865405&{}:{}&0.211324865405&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.211324865405&{}:{}&0.577350269190&{}:{}&0.211324865405&,\\C^\prime_{\mathbf{6}}&{}\approx{}&0.981125224325&{}:{}&0.981125224325&{}:{}&-0.962250448649&. \end{alignedat} \]
6 (220)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6}}&{}\approx{}&0.498714618890&{}:{}&0.498714618890&{}:{}&0.002570762220&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.347748149397&{}:{}&0.648906880894&{}:{}&0.003344969709&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.648906880894&{}:{}&0.347748149397&{}:{}&0.003344969709&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.394337567297&{}:{}&0.394337567297&{}:{}&0.211324865405&. \end{alignedat} \]
6 (220)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6}}&{}\approx{}&0.609378523886&{}:{}&0.609378523886&{}:{}&-0.218757047773&,\\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.363466037631&{}:{}&0.993007681634&{}:{}&-0.356473719264&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.993007681634&{}:{}&0.363466037631&{}:{}&-0.356473719264&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.422649730810&{}:{}&0.422649730810&{}:{}&0.154700538379&. \end{alignedat} \]
6 (220)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6}}&{}\approx{}&0.451387728771&{}:{}&0.451387728771&{}:{}&0.097224542457&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.577350269190&{}:{}&0.211324865405&{}:{}&0.211324865405&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.211324865405&{}:{}&0.577350269190&{}:{}&0.211324865405&,\\C^\prime_{\mathbf{6}}&{}\approx{}&0.981125224325&{}:{}&0.981125224325&{}:{}&-0.962250448649&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.347748149397&{}:{}&0.648906880894&{}:{}&0.003344969709&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.648906880894&{}:{}&0.347748149397&{}:{}&0.003344969709&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.394337567297&{}:{}&0.394337567297&{}:{}&0.211324865405&, \end{alignedat} \]
6 (220)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6}}&{}\approx{}&0.518617531751&{}:{}&0.518617531751&{}:{}&-0.037235063502&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.077350269190&{}:{}&-0.077350269190&,\\B^*_{\mathbf{6}}&{}\approx{}&1.077350269190&{}:{}&0.000000000000&{}:{}&-0.077350269190&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6}}&{}\approx{}&0.482672825160&{}:{}&0.482672825160&{}:{}&0.034654349680&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.347748149397&{}:{}&0.648906880894&{}:{}&0.003344969709&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.648906880894&{}:{}&0.347748149397&{}:{}&0.003344969709&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.394337567297&{}:{}&0.394337567297&{}:{}&0.211324865405&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.077350269190&{}:{}&-0.077350269190&,\\B^*_{\mathbf{6}}&{}\approx{}&1.077350269190&{}:{}&0.000000000000&{}:{}&-0.077350269190&,\\C^*_{\mathbf{6}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)